prove that if x,y,z is a Pythagorean triple, than at least one of x,y,z is divisible by 5.

What I have so far: let $\displaystyle x = r^2 - s^2, y = 2rs, z = r^2 + s^2 $
Then try out all of the cases. If $\displaystyle 5\nmid x \mbox{ and } 5\nmid y,\mbox{ then } 5\mid z$ and use congruences to show that it is true, for all of the possible values of r and s.

Is there a more elegant way to prove this, because what I have so far is too tedious.

Nov 23rd 2009, 08:15 PM

CaptainBlack

Quote:

Originally Posted by keityo

prove that if x,y,z is a Pythagorean triple, than at least one of x,y,z is divisible by 5.

What I have so far: let $\displaystyle x = r^2 - s^2, y = 2rs, z = r^2 + s^2 $
Then try out all of the cases. If $\displaystyle 5\nmid x \mbox{ and } 5\nmid y,\mbox{ then } 5\mid z$ and use congruences to show that it is true, for all of the possible values of r and s.

Is there a more elegant way to prove this, because what I have so far is too tedious.

Consider the possible values of squares modulo $\displaystyle 5$, then all the possible values of $\displaystyle x^2+y^2$ modulo $\displaystyle 5$ and which of these can a square modulo 5.